# Hybrid Chebyshev function bases for sparse spectral methods in parity-mixed PDEs on an infinite domain

## TL;DR

This paper introduces a spectral method using hybrid Chebyshev function bases to efficiently solve differential equations on infinite domains, maintaining sparsity even with parity-mixed operators, enabling fast and accurate computations.

## Contribution

The paper develops a novel hybrid Chebyshev rational function basis that preserves sparsity in spectral discretizations of parity-mixed PDEs on infinite domains.

## Key findings

- Achieves $O(N \, \ln N)$ computational complexity.
- Maintains spectral accuracy in unbounded geometries.
- Enables fast implicit-explicit time-marching schemes.

## Abstract

We present a numerical spectral method to solve systems of differential equations on an infinite interval $y\in (-\infty, \infty)$ in presence of linear differential operators of the form $Q(y) \left(\partial/\partial_y\right)^b$ (where $Q(y)$ is a rational fraction and $b$ a positive integer). Even when these operators are not parity-preserving, we demonstrate how a mixed expansion in interleaved Chebyshev rational functions $TB_n(y)$ and $SB_n(y)$ preserves the sparsity of their discretization. This paves the way for fast $O(N\ln N)$ and spectrally accurate mixed implicit-explicit time-marching of sets of linear and nonlinear equations in unbounded geometries.